Time Value of Money
Download
Report
Transcript Time Value of Money
Chapter 4
Time Value of Money
1
Time Value Topics
Future value
Present value
Rates of return
Amortization
2
Determinants of Intrinsic Value:
The Present Value Equation
Net operating
profit after taxes
Free cash flow
(FCF)
Value =
Required investments
in operating capital
−
=
FCF1
FCF2
FCF∞
... +
+
+
(1 + WACC)1
(1 + WACC)2
(1 + WACC)∞
Weighted average
cost of capital
(WACC)
Market interest rates
Cost of debt
Firm’s debt/equity mix
Market risk aversion
Cost of equity
Firm’s business risk
Time lines show timing of cash
flows.
0
1
2
3
CF1
CF2
CF3
I%
CF0
Tick marks at ends of periods, so Time 0
is today; Time 1 is the end of Period 1; or
the beginning of Period 2.
4
Time line for a $100 lump sum
due at the end of Year 2.
0
I%
1
2 Year
100
5
Time line for an ordinary
annuity of $100 for 3 years
0
I%
1
2
3
100
100
100
6
Time line for uneven CFs
0
-50
I%
1
2
3
100
75
50
7
FV of an initial $100 after
3 years (I = 10%)
0
1
2
3
10%
100
FV = ?
Finding FVs (moving to the right
on a time line) is called compounding.
8
After 1 year
FV1 =
=
=
=
PV + INT1 = PV + PV (I)
PV(1 + I)
$100(1.10)
$110.00
9
After 2 years
FV2 =
=
=
=
FV1(1+I) = PV(1 + I)(1+I)
PV(1+I)2
$100(1.10)2
$121.00
10
After 3 years
FV3 =
=
=
=
FV2(1+I)=PV(1 + I)2(1+I)
PV(1+I)3
$100(1.10)3
$133.10
In general,
FVN = PV(1 + I)N
11
Four Ways to Find FVs
Step-by-step approach using time line
(as shown in Slides 7-10).
Solve the equation with a regular
calculator (formula approach).
Use a financial calculator.
Use a spreadsheet.
12
Financial calculator: HP10BII
Adjust display brightness: hold down
ON and push + or –.
Set number of decimal places to
display: Orange Shift key, then DISP
key (in orange), then desired decimal
places (e.g., 3).
To temporarily show all digits, hit
Orange Shift key, then DISP, then =.
13
HP10BII (Continued)
To permanently show all digits, hit
ORANGE shift, then DISP, then . (period
key).
Set decimal mode: Hit ORANGE shift,
then ./, key. Note: many non-US
countries reverse the US use of
decimals and commas when writing a
number.
14
HP10BII: Set Time Value
Parameters
To set END (for cash flows occurring at
the end of the year), hit ORANGE shift
key, then BEG/END.
To set 1 payment per period, hit 1, then
ORANGE shift key, then P/YR.
15
Financial Calculator Solution
Financial calculators solve this
equation:
FVN + PV (1+I)N = 0.
There are 4 variables. If 3 are
known, the calculator will solve for
the 4th.
16
Here’s the setup to find FV
INPUTS
3
N
10
-100
I/YR PV
0
PMT
OUTPUT
FV
133.10
Clearing automatically sets everything to 0,
but for safety enter PMT = 0.
Set: P/YR = 1, END.
17
Spreadsheet Solution
Use the FV function: see spreadsheet in
Ch04 Mini Case.xls
= FV(I, N, PMT, PV)
= FV(0.10, 3, 0, -100) = 133.10
18
What’s the PV of $100 due in
3 years if I/YR = 10%?
Finding PVs is discounting, and it’s the
reverse of compounding.
0
PV = ?
10%
1
2
3
100
19
Solve FVN = PV(1 + I )N for PV
PV =
FVN
(1+I)N
= FVN
1
PV = $100
1.10
1
1+I
N
3
= $100(0.7513) = $75.13
20
Financial Calculator Solution
INPUTS
OUTPUT
3
N
10
I/YR
PV
-75.13
0
PMT
100
FV
Either PV or FV must be negative. Here
PV = -75.13. Put in $75.13 today, take
out $100 after 3 years.
21
Spreadsheet Solution
Use the PV function: see spreadsheet in
Ch04 Mini Case.xls
= PV(I, N, PMT, FV)
= PV(0.10, 3, 0, 100) = -75.13
22
Finding the Time to Double
0
-1
20%
1
2
FV = PV(1 + I)N
?
2
Continued on next slide
23
Time to Double (Continued)
$2
(1.2)N
N LN(1.2)
N
N
=
=
=
=
=
$1(1 + 0.20)N
$2/$1 = 2
LN(2)
LN(2)/LN(1.2)
0.693/0.182 = 3.8
24
Financial Calculator Solution
INPUTS
N
OUTPUT 3.8
20
I/YR
-1
PV
0
PMT
2
FV
25
Spreadsheet Solution
Use the NPER function: see spreadsheet
in Ch04 Mini Case.xls
= NPER(I, PMT, PV, FV)
= NPER(0.10, 0, -1, 2) = 3.8
26
0
-1
?%
1
FV =
$2 =
(2)(1/3) =
1.2599 =
I=
2
I)N
PV(1 +
$1(1 + I)3
(1 + I)
(1 + I)
0.2599 = 25.99%
3
2
27
Financial Calculator
INPUTS
OUTPUT
3
N
I/YR
25.99
-1
PV
0
PMT
2
FV
28
Spreadsheet Solution
Use the RATE function:
= RATE(N, PMT, PV, FV)
= RATE(3, 0, -1, 2) = 0.2599
29
Ordinary Annuity vs. Annuity
Due
Ordinary Annuity
0
I%
1
2
3
PMT
PMT
PMT
1
2
3
PMT
PMT
Annuity Due
0
PMT
I%
30
What’s the FV of a 3-year
ordinary annuity of $100 at 10%?
0
10%
1
2
100
100
3
FV
100
110
121
= 331
31
FV Annuity Formula
The future value of an annuity with N
periods and an interest rate of I can be
found with the following formula:
= PMT
(1+I)N-1
I
= $100
(1+0.10)3-1
0.10
= $331
32
Financial Calculator Formula
for Annuities
Financial calculators solve this equation:
FVN + PV(1+I)N + PMT
(1+I)N-1
I
=0
There are 5 variables. If 4 are known,
the calculator will solve for the 5th.
33
Financial Calculator Solution
INPUTS
OUTPUT
3
10
0
-100
N
I/YR
PV
PMT
FV
331.00
Have payments but no lump sum PV, so
enter 0 for present value.
34
Spreadsheet Solution
Use the FV function: see spreadsheet.
= FV(I, N, PMT, PV)
= FV(0.10, 3, -100, 0) = 331.00
35
What’s the PV of this ordinary
annuity?
0
1
2
3
100
100
100
10%
90.91
82.64
75.13
248.69 = PV
36
PV Annuity Formula
The present value of an annuity with N
periods and an interest rate of I can be
found with the following formula:
= PMT
1
I
−
1
I (1+I)N
1
1
= $100
−
0.1
0.1(1+0.1)3
= $248.69
37
Financial Calculator Solution
INPUTS
OUTPUT
3
10
N
I/YR
PV
100
0
PMT
FV
-248.69
Have payments but no lump sum FV, so
enter 0 for future value.
38
Spreadsheet Solution
Use the PV function: see spreadsheet.
= PV(I, N, PMT, FV)
= PV(0.10, 3, 100, 0) = -248.69
39
Find the FV and PV if the
annuity were an annuity due.
0
1
2
100
100
3
10%
100
40
PV and FV of Annuity Due
vs. Ordinary Annuity
PV of annuity due:
= (PV of ordinary annuity) (1+I)
= ($248.69) (1+ 0.10) = $273.56
FV of annuity due:
= (FV of ordinary annuity) (1+I)
= ($331.00) (1+ 0.10) = $364.10
41
PV of Annuity Due: Switch
from “End” to “Begin”
BEGIN Mode
INPUTS
OUTPUT
3
10
N
I/YR
PV
100
0
PMT
FV
-273.55
42
FV of Annuity Due: Switch
from “End” to “Begin”
BEGIN Mode
INPUTS
OUTPUT
3
10
N
I/YR
0
100
PV
PMT
FV
-364.10
43
Excel Function for Annuities
Due
Change the formula to:
=PV(0.10,3,-100,0,1)
The fourth term, 0, tells the function there
are no other cash flows. The fifth term tells
the function that it is an annuity due. A
similar function gives the future value of an
annuity due:
=FV(0.10,3,-100,0,1)
44
What is the PV of this
uneven cash flow stream?
0
90.91
247.93
225.39
-34.15
1
2
3
4
100
300
300
-50
10%
530.08 = PV
45
Financial calculator: HP10BII
Clear all: Orange Shift key, then C All
key (in orange).
Enter number, then hit the CFj key.
Repeat for all cash flows, in order.
To find NPV: Enter interest rate (I/YR).
Then Orange Shift key, then NPV key
(in orange).
46
Financial calculator: HP10BII
(more)
To see current cash flow in list, hit RCL
CFj CFj
To see previous CF, hit RCL CFj –
To see subsequent CF, hit RCL CFj +
To see CF 0-9, hit RCL CFj 1 (to see CF 1).
To see CF 10-14, hit RCL CFj . (period) 1
(to see CF 11).
47
Input in “CFLO” register:
CF0
CF1
CF2
CF3
CF4
=
=
=
=
=
0
100
300
300
-50
Enter I/YR = 10, then press NPV button
to get NPV = 530.09. (Here NPV = PV.)
48
Excel Formula in cell A3:
=NPV(10%,B2:E2)
49
Nominal rate (INOM)
Stated in contracts, and quoted by
banks and brokers.
Not used in calculations or shown on
time lines
Periods per year (M) must be given.
Examples:
8%; Quarterly
8%, Daily interest (365 days)
50
Periodic rate (IPER )
IPER = INOM/M, where M is number of compounding
periods per year. M = 4 for quarterly, 12 for monthly,
and 360 or 365 for daily compounding.
Used in calculations, shown on time lines.
Examples:
8% quarterly: IPER = 8%/4 = 2%.
8% daily (365): IPER = 8%/365 = 0.021918%.
51
The Impact of Compounding
Will the FV of a lump sum be larger or
smaller if we compound more often,
holding the stated I% constant?
Why?
52
The Impact of Compounding
(Answer)
LARGER!
If compounding is more frequent than
once a year--for example, semiannually,
quarterly, or daily--interest is earned on
interest more often.
53
FV Formula with Different
Compounding Periods
INOM
FVN = PV 1 +
M
MN
54
$100 at a 12% nominal rate with
semiannual compounding for 5 years
INOM
FVN = PV 1 +
M
FV5S
0.12
= $100 1 +
2
= $100(1.06)10
MN
2x5
= $179.08
55
FV of $100 at a 12% nominal rate for
5 years with different compounding
FV(Ann.)
FV(Semi.)
FV(Quar.)
FV(Mon.)
FV(Daily)
=
=
=
=
=
$100(1.12)5
$100(1.06)10
$100(1.03)20
$100(1.01)60
$100(1+(0.12/365))(5x365)
=
=
=
=
=
$176.23
$179.08
$180.61
$181.67
$182.19
56
Effective Annual Rate (EAR =
EFF%)
The EAR is the annual rate that causes
PV to grow to the same FV as under
multi-period compounding.
57
Effective Annual Rate Example
Example: Invest $1 for one year at 12%,
semiannual:
FV = PV(1 + INOM/M)M
FV = $1 (1.06)2 = $1.1236.
EFF% = 12.36%, because $1 invested for
one year at 12% semiannual compounding
would grow to the same value as $1 invested
for one year at 12.36% annual compounding.
58
Comparing Rates
An investment with monthly payments
is different from one with quarterly
payments. Must put on EFF% basis to
compare rates of return. Use EFF%
only for comparisons.
Banks say “interest paid daily.” Same
as compounded daily.
59
EFF% for a nominal rate of 12%,
compounded semiannually
EFF% =
=
INOM
1 +
M
0.12
1 +
2
M
−1
2
−1
= (1.06)2 - 1.0
= 0.1236 = 12.36%.
60
Finding EFF with HP10BII
Type in nominal rate, then Orange Shift
key, then NOM% key (in orange).
Type in number of periods, then Orange
Shift key, then P/YR key (in orange).
To find effective rate, hit Orange Shift
key, then EFF% key (in orange).
61
EAR (or EFF%) for a Nominal
Rate of of 12%
EARAnnual
= 12%.
EARQ
= (1 + 0.12/4)4 - 1
= 12.55%.
EARM
= (1 + 0.12/12)12 - 1 = 12.68%.
EARD(365)
= (1 + 0.12/365)365 - 1= 12.75%.
62
Can the effective rate ever be
equal to the nominal rate?
Yes, but only if annual compounding is
used, i.e., if M = 1.
If M > 1, EFF% will always be greater
than the nominal rate.
63
When is each rate used?
INOM:
Written into contracts, quoted
by banks and brokers. Not used
in calculations or shown
on time lines.
64
When is each rate used?
(Continued)
IPER: Used in calculations, shown on
time lines.
If INOM has annual compounding,
then IPER = INOM/1 = INOM.
65
When is each rate used?
(Continued)
EAR (or EFF%): Used to compare
returns on investments with different
payments per year.
Used for calculations if and only if
dealing with annuities where payments
don’t match interest compounding
periods.
66
Amortization
Construct an amortization schedule for
a $1,000, 10% annual rate loan with 3
equal payments.
67
Step 1: Find the required
payments.
0
10%
-1,000
INPUTS
OUTPUT
3
N
1
2
3
PMT
PMT
PMT
10
I/Y
R
-1000
PV
0
PMT
FV
402.11
68
Step 2: Find interest charge
for Year 1.
INTt = Beg balt (I)
INT1 = $1,000(0.10) = $100
69
Step 3: Find repayment of
principal in Year 1.
Repmt = PMT - INT
= $402.11 - $100
= $302.11
70
Step 4: Find ending balance
after Year 1.
End bal = Beg bal - Repmt
= $1,000 - $302.11 = $697.89
Repeat these steps for Years 2 and 3
to complete the amortization table.
71
Amortization Table
BEG
BAL
$1,000
PMT
$402
2
698
402
70
332
366
3
366
402
37
366
0
YEAR
1
TOT
1,206.34
PRIN
INT
PMT
$100 $302
END
BAL
$698
206.34 1,000
72
Interest declines because
outstanding balance declines.
$450
$400
$350
$300
$250
$200
$150
$100
$50
$0
Interest
Principal
PMT 1
PMT 2
PMT 3
73
Amortization tables are widely
used--for home mortgages, auto
loans, business loans, retirement
plans, and more. They are very
important!
Financial calculators (and
spreadsheets) are great for setting
up amortization tables.
74
Fractional Time Periods
On January 1 you deposit $100 in an
account that pays a nominal interest
rate of 11.33463%, with daily
compounding (365 days).
How much will you have on October 1,
or after 9 months (273 days)? (Days
given.)
75
Convert interest to daily rate
IPER = 11.33463%/365
= 0.031054% per day
0
1
2
273
0.031054%
-100
FV=?
76
Find FV
FV273 = $100 (1.00031054)273
= $100 (1.08846) = $108.85
77
Calculator Solution
IPER = INOM/M
= 11.33463/365
= 0.031054 per day.
INPUTS
273
N
OUTPUT
I/YR
-100
0
PV
PMT
FV
108.85
78
Non-matching rates and periods
What’s the value at the end of Year 3 of
the following CF stream if the quoted
interest rate is 10%, compounded
semiannually?
79
Time line for non-matching
rates and periods
0
1
2
3
4
5%
100
100
5
6
6-mos.
periods
100
80
Non-matching rates and periods
Payments occur annually, but
compounding occurs each 6 months.
So we can’t use normal annuity
valuation techniques.
81
1st Method: Compound Each
CF
0
5%
1
2
100
3
4
100
5
6
100.00
110.25
121.55
331.80
FVA3 = $100(1.05)4 + $100(1.05)2 + $100
= $331.80
82
2nd Method: Treat as an
annuity, use financial calculator
Find the EFF% (EAR) for the quoted rate:
EFF% =
0.10
1 +
2
2
− 1 = 10.25%
83
Use EAR = 10.25% as the
annual rate in calculator.
INPUTS
OUTPUT
3
10.25
0
-100
N
I/YR
PV
PMT
FV
331.80
84
What’s the PV of this stream?
0
5%
90.70
82.27
74.62
247.59
1
2
3
100
100
100
85
Comparing Investments
You are offered a note that pays
$1,000 in 15 months (or 456 days) for
$850. You have $850 in a bank that
pays a 6.76649% nominal rate, with
365 daily compounding, which is a daily
rate of 0.018538% and an EAR of
7.0%. You plan to leave the money in
the bank if you don’t buy the note.
The note is riskless.
Should you buy it?
86
Daily time line
IPER =
0
-850
0.018538% per day.
…
365
…
456 days
1,000
87
Three solution methods
1. Greatest future wealth: FV
2. Greatest wealth today: PV
3. Highest rate of return: EFF%
88
1. Greatest Future Wealth
Find FV of $850 left in bank for
15 months and compare with
note’s FV = $1,000.
FVBank
=
=
$850(1.00018538)456
$924.97 in bank.
Buy the note: $1,000 > $924.97.
89
Calculator Solution to FV
IPER = INOM/M
= 6.76649/365
= 0.018538 per day.
INPUTS
456
N
OUTPUT
I/YR
-850
0
PV
PMT
FV
924.97
90
2. Greatest Present Wealth
Find PV of note, and compare
with its $850 cost:
PV =
=
$1,000/(1.00018538)456
$918.95
Buy the note: $918.95 > $850
91
Financial Calculator Solution
INPUTS
6.76649/365 =
456 .018538
N
OUTPUT
I/YR
PV
0
1000
PMT
FV
-918.95
PV of note is greater than its $850
cost, so buy the note. Raises your
wealth.
92
3. Rate of Return
Find the EFF% on note and compare
with 7.0% bank pays, which is your
opportunity cost of capital:
FVN = PV(1 + I)N
$1,000 = $850(1 + I)456
Now we must solve for I.
93
Calculator Solution
INPUTS
456
N
OUTPUT
-850
I/YR
PV
0.035646%
per day
0
1000
PMT
FV
Convert % to decimal:
Decimal = 0.035646/100 = 0.00035646.
EAR = EFF% = (1.00035646)365 - 1
= 13.89%.
94
Using interest conversion
P/YR
= 365
NOM% = 0.035646(365) = 13.01
EFF% = 13.89
Since 13.89% > 7.0% opportunity cost,
buy the note.
95